This article covers methods to tackle AWS "ERROR 2003 (HY000): Can't connect to MySQL server" for our customers.
You can easily fix this issue by adding the public ip address of my EC2 server to the DB security group default policy.
You can create a new security group with the corresponding IP range/subnet of your EC2 instances(s). You can use the exact IP(s), or use your VPC's subnet(s) as the range.
Once created, you can right click your RDS instance in the management console and select 'Modify'. From there, you can change the associated security groups.